Re: hot air rising

[John Denker <jsd@MONMOUTH.COM>]

At 10:22 AM 7/21/99 -0400, Robert A Cohen wrote:
> > > On Tue, 20 Jul 1999, Doug Craigen wrote:
> > >
> > > > One of my  examples to show how people learn particular situations as
> > > > absolute rules - I ask how it is that hot air rises but the air is
> > > > colder the higher up you go in the atmosphere (where all the hot air
> > > > went)....  most people are at a loss, having never even realized a
> > > > contradiction between two of their absolute rules.
...
> Now that I know about the "source of heat" theory, I must say that I don't
> have a good answer to this.  The "close to magma" theory can be somewhat
> addressed by measuring the temperature below ground (not in mines, mind
> you).  The "close to ground" theory can be somewhat addressed by measuring
> the temperature on mountain tops, as you say.  Neither totally convinces
> the students of the fallacy of their theory (is Denver always colder than
> New York City?) but they do get the students to question.  I then
> introduce another "theory", based upon mixing, and ask the students how
> they might determine the "better" theory.

I'm glad your students are not convinced by the "source of heat" or
"mixing" theories because both of those miss the essential physical points,
namely:

 a) Air tends to rise if it is hotter THAN THE SURROUNDING AIR (other
things like humidity being equal).
 b) Air that goes up for any reason (including wind forced up the slope of
a mountain) will COOL ADIABATICALLY (to a good approximation) as it rises.

Energy is conserved.  But heat is not the only form of energy in an air
parcel.  When it rises it expands, doing work against the pressure of the
surrounding air.  If you carry the analysis to the next level of detail,
the air pressure is attributable to gravity, so air is (more or less
indirectly, depending on your point of view) doing work against the
gravitational field.  The air loses temperature as surely as a roller
coaster loses kinetic energy when it goes up the hill.

After a parcel of air has risen, it will be cooler than it was before it
rose.  It may or may not be cooler than the surrounding air, so it may or
may not continue to rise.  On a typical fair-weather day (e.g. the day
after a cold front has passed through) you can demonstrate this pretty
convincingly in a light plane.  There will be a height below which the air
will be bumpy, and above which it will be smooth as glass.  This height
depends on the temperature-versus-height profile of the air, and on the
temperature of the ground.  (Remember that the ground is heated by the sun;
 the sun has almost no direct effect on the air.)  This magic height is the
height to which a parcel of air, initially heated by the sunny ground,
rises before it is no longer warmer than the surrounding air.

If the air initially contains moisture, it may happen that it cools to the
dewpoint before it stops rising, in which case it gets an additional "kick"
from the latent heat of condensation.

BTW, the height of the troposphere itself is determined (for all practical
purposes) by the height of the tallest thunderstorms, i.e. the height to
which warm moist air can rise before it is no longer warmer than the
surrounding air and stops rising.  Below this altitude, the
temperature-versus-height profile is approximately adiabatic, because of
all this rising and descending air.  Above this height, we have the
stratosphere, which is isothermal (to a good approximation).  Thermal
equilibrium is exactly what everybody would expect for a chunk of gas that
was *not* subjected to all these adiabatic updrafts and downdrafts.